IE 300/GE 331 Lecture 17
Negar Kiyavash, UIUC
1
•
Recall: Normal distribution:
•
Let R.V.s U and V be independent normal, what can
we say about X=aU+bV and Y=cU+dV?
(a, b, c, d
are scalars)
•
They are Normal too, but not independent.
•
In fact any
linear combination
of X and Y is normal!
Bivariate Normal
Distribution
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
=
2
2
2
)
(
exp
2
1
)
(
σ
μ
σ
π
x
x
f
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IE 300/GE 331 Lecture 17
Negar Kiyavash, UIUC
2
Bivariate Normal
Distribution
•
U and V are independent normal, X=aU+bV and
Y=cU+dV
(a, b, c, d are scalars)
•
X,Y are
jointly normal
(Gaussian) (Attention
jointly
normal
is not same as normal!)
•
The Joint PDF is called a
bivariate normal
distribution:
with
⎟
⎠
⎞
⎜
⎝
⎛
−
−
−
=
−
)
(
)
(
2
1
exp


2
1
)
(
1
T
μ
x
Σ
μ
x
Σ
x
π
f
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
2
2
,
,
y
y
x
y
x
x
Y
X
y
x
σ
σ
ρσ
σ
ρσ
σ
μ
μ
Σ
μ
x